Journal of Applied Mechanics Vol.12, pp.719-728　（August 2009）
Journal of Applied Mechanics Vol.12 (August 2009)
JSCE
JSCE
Influence of Mean Step Length on Sand Dune Dynamics:
Nonuniform Sediment Case
Wandee THAISIAM1, Yasuyuki SHIMIZU2 and Ichiro KIMURA3
1Student Member of JSCE, Doctoral Student, Dept. of Hydraulic Research, Hokkaido University (North
13, West 8, Kita-ku, Sapporo 060-8628, Japan)
2Member of JSCE, Dr. Eng., Professor, Dept. of Hydraulic Research, Hokkaido University (North 13,
West 8, Kita-ku, Sapporo 060-8628, Japan)
3Member of JSCE, Dr. Eng., Associate Professor, Dept. of Hydraulic Research, Hokkaido University
(North 13, West 8, Kita-ku, Sapporo 060-8628, Japan)
This work presents the sand dune study using the morphodynamic numerical
model for nonuniform sediment. We assess the influence of mean step length, an
important parameter used in sediment transport formula proposed by Tsujimoto
and Motohashi, on sand dune dynamics. The equilibrium bed load transport
formula is also employed in order to compare the results using different sediment
transport formulae. From the results of the nonequilibrium sediment transport
model, we found that the dune geometry significantly depends on the mean step
length, and the appropriate value of mean step length for the simulation of
nonuniform sediment is found to be 20-30 times sediment diameter. The results
using the nonequilibrium sediment transport model provide better agreement with
the experiments than that using the equilibrium sediment transport model.
Moreover, the model sensitivity assessment, i.e. the initial perturbation and the
domain length is conducted.
Key Words: mean step length, nonuniform sediment, dune formation
1. Introduction
Prediction of morphodynamic in alluvial rivers is still
challenging problem in the field of river engineering.
Since the bed material is composed of many sizes of
grain, the bedform geometry and their evolutions are too
complicated for modeling. The past studies model the
bedform phenomena by using median size of sediment.
As the results, some important phenomena such as
bedform geometry and their migration cannot be
explained sufficiently. In order to well understand the
bedform characteristics of nonuniform sediment, both
experimental study and numerical model study of
nonuniform sediment are still needed.
The morphodynamic numerical model of nonuniform
sediment recently introduces to the filed of river
engineering. Most of them initiated the nonuniform
sediment transport approach in the meso-scale bedform
simulations. Bui and Rutschmann1) performed a 3D
numerical model of nonuniform sediment transport in
alluvial channel. In the calculation codes, the flow was
solved by using full Reynolds-averaged Navier-Strokes
equations with k-ε turbulence model.
While the
sediment transport model was calculated by using a
nonequilibrium sediment transport model. Fractional
sediment transport and a multiple layer model were
employed for sediment sorting calculation. The main
features of grain sorting and bed deformation in the
channel scale simulation were generally well reproduced.
The model results were found to be depended on the
value of the nonequilibrium adaptation length. With the
use of the nonequilibrium adaptation length formula
proposed by Phillips and Sutherland2), the model results
agreed best with the measurement. Wu3) proposed a
depth-averaged two-dimensional numerical modeling for
unsteady flow and nonuniform sediment transport in open
channels. The 2D shallow water equations were solved
by the SIMPLE(C) algorithms with the Rhie and Chow’s
momentum interpolation technique.
The sediment
transport model adopted a nonequilibrium approach for
nonuniform total load sediment transport and the
nonequilibrium adaptation length was set to be related to
length of sand dune and alternate bars. Although, the
- 719 -
sediment module adopted a coupling procedure for the
computations of sediment transport, bed deformation,
sediment sorting, but flow model and sediment transport
model were simulated in a decoupled manner. The
model showed good agreement with the measured data in
meso-scale bedforms. By the use of the nonequilibrium
sediment transport model, some recent studies1),3) showed
that the model seemed adequately to reproduce the bed
deformation and sediment sorting in the meso-scale
bedforms. However, these model results were found to
be sensitive with the value of the mean step length.
Not only nonequilibrium sediment transport model is
widely employed in the field of numerical simulation, but
the equilibrium sediment transport model is also familiar
in this field.
Sloff et al.4) performed a 2D
morphodynamic modeling with nonuniform sediment in
order to simulate typical morphological behavior for
rivers characterized by nonuniform sediment. The
equilibrium sediment transport model and the bed layer
model were employed in the sediment transport process.
The model was tested with a curved flume experimental
data and the Tenryuu River observed data. The model
results showed good agreement with the experiments.
With the complex situations of Tenyuu River, the
simulation results were consistent with the observations.
Ashida et al.5) worked on a numerical calculation of flow
and bed evolution in compound channels using a
two-layer flow model. The model was constructed to
deal with nonuniform sediment in order to evaluate the
effects of sediment sorting on bed deformation. The
concept of bed layer model was proposed for grain
sorting calculation. The equilibrium sediment transport
approach was used in sediment transport process.
Comparing the simulated results with the observed data,
the model had ability to simulate bed deformation and
sediment sorting phenomenon in meandering channel
with flood plain flow. From previous studies, it was
shown that the numerical model provided a good
performance on bedform evolution and sediment sorting
in the river scale simulation by using the equilibrium
sediment transport approach.
Recently, the numerical model for micro-scale
bedform simulation of nonuniform sediment was
proposed by Thaisiam et al.6) for simulating sand dune
characteristics and grain sorting inside sand dunes. In
the numerical model, the sediment transport model
explicitly takes the turbulent flow model into account
during the morphodynamic computation. The sediment
transport model was composed of a nonuniform sediment
transport model and a bed layer model. The concept of
size fraction transport and a nonequilibrium bed load
transport approach were employed in the nonuniform
sediment transport model. The bed layer model was
applied for grain sorting simulation inside sand dunes.
In addition, the hydrodynamic model proposed by Giri
and Shimizu7) was used for hydrodynamic simulation.
This model was able to simulate the morphodynamic
features and grain sorting inside sand dunes in a
physically based manner.
In the present study, we perform numerical studies in
order to assess the influence of mean step length on sand
dune dynamics. Two different approaches of mean step
length are employed in the calculation. The equilibrium
bed load transport formula is used to evaluate the
performance of our morphodynamic model. In addition,
the model sensitivity assessment is conducted in order to
test our numerical model for nonuniform sediment with
some factors. The numerical model has been tested with
different domain length and initial perturbation patterns
on bed.
2. Hydrodynamic model
The
governing
equations
for
unsteady
two-dimensional flow in the Cartesian coordinate system
(x,y,t) is transformed to the moving boundary fitted
coordinate system (ξ, η, τ)8).
The transformed
equations are solved by splitting into a non-advection and
a pure advection term. The non-advection term is
solved by using central difference method. The pure
advection term is calculated by using a high-order
Godunov scheme known as the cubic interpolated
pseudoparticle (CIP) technique. The pressure term is
resolved using SOR method.
In the turbulence model, a 2nd order non-linear k-ε
turbulence closure is employed to reproduce turbulent
characteristics in shear flow with separation zone. From
the past study7,9), the 2nd order non-linear k-ε turbulence
closure satisfactorily predicts mean flow and turbulent
properties over the bedforms.
The time-dependent water surface change computation
is used for realistic reproduction of free surface flow over
migrating bedforms. The model is able to accomplish
stable and reasonable solutions with a free surface flow
condition over the migrating bed forms. The kinematic
condition is established along the water surface in order
to compute water surface variation.
The boundary condition at the bed is no slip. A
logarithmic expression for near-bed region is adopted.
Although the logarithmic expression is found to be an
inappropriate method for simulating the flow at the
boundary condition with flow separation, Giri and
Shimizu7) employed the logarithmic expression at the
boundary condition with flow separation to simulate the
flow over the fixed dune bed. From their study, the time
averaged streamwise and vertical velocity profiles in all
regions over dunes were reproduced reasonably well,
although some discrepancies were noticed with respect to
streamwise normal stress, particularly in separation
region behind the dune crest.
However, the
sophisticated model for calculating flow in the separation
zone over the bedform migration is needed in the further
study. The periodic boundary condition is employed in
the computation domain in which output at the
downstream end is set to be input at the upstream end.
3. Sediment transport model
The sediment transport model for nonuniform
sediment was proposed for bedform evolution and grain
- 720 -
sorting simulations. We used both the concept of size
fraction and bed load transport model of nonuniform
sediment for calculating in the sediment exchange
process. The grain sorting simulation was performed by
using the bed layer model. The sediment transport
model can be then divided into two submodels which are
a nonuniform sediment transport and a bed layer model.
pickup rate formula for nonuniform sediment by using
the critical tractive force for each sediment size fraction.
The pickup rate for sediment size fraction k can be
written as follow:
3.1 Nonuniform sediment transport model
The concept of size fraction transport is to divide bed
material into size fractions which consider each size
fraction as a uniform material. The bed material
transport rate can be calculated by multiplying the
potential transport rate corresponding to a given size
fraction with the percentage of material which can be
read as follows:
where psk is the pickup rate of sediment size fraction k, ρ
and ρs are fluid and sediment density respectively, τ*k is
dimensionless local bed shear stress of sediment size
fraction k, and τ*ck is dimensionless critical bed shear
stress of sediment size fraction k.
The sediment deposition rate is expressed as:
nk
nk
qb = ∑ qbk =∑ Pk .qk
k =1
(1)
k =1
where qb is the bed load transport rate per unit width, qbk
is the bed load transport of sediment size fraction k per
unit width, Pk is the concentration of sediment size
fraction k, qk is the potential transport rate for a given size
fraction k, subscripts k and nk are the number and the
total number of size fraction respectively.
In the present study, we use both the equilibrium bed
load transport and nonequilibrium bed load transport
approaches in our sediment transport module. We
perform numerical calculations in both sediment transport
approaches in order to evaluate the model performance
on bedform evolution simulations. Thus, the brief
description of both approaches can be described as
follows.
(1) Nonequilibrium bed load transport approach
The nonequilibrium sediment transport model has
been widely introduced in micro-scale bedform
simulation of uniform sediment. For instant, Giri and
Shimizu6) used the nonequilibrium sediment transport
approach in a vertical two-dimensional morphodynamic
model for calculating sediment pickup and deposition
rate in the sediment exchange process. The proposed
model was found to be in good agreement with
experimental data in term of bedform geometry
prediction. Onda and Hosoda10) proposed a depth
averaged flow model combined with a nonequilibrium
sediment transport model. They succeed to reproduce
the formation process of micro-scale sand waves and the
shape characteristics of dune.
By using the
nonequilibrium sediment transport approach in the
computation model, it seems adequately to reproduce
bedform evolution of uniform sediment.
For uniform sediment, Nakagawa and Tsujimoto11)
proposed a nonequilibrium transport model characterized
by pickup rate and step length, and applied it successfully
to explain several alluvial processes such as the
micro-scale bed deformation. Based on the uniform
material concept, Tsujimoto and Motohashi12) advanced a
(
0.03τ *k 1 − 0.7τ *ck / τ *k
psk =
d k /( ρ s / ρ − 1) g
)
3
pdk = ∫ psk f s ( s )ds
(2)
(3)
where pdk is the deposition rate of sediment size fraction k
and fs(s) is the distribution function of step length.
The exponential distribution function of step length of
the kth size fraction of bed material by Nakagawa and
Tsujimoto11) applied for nonuniform sediment is written
as:
f sk ( s ) =
1
Λ
⎛ s⎞
exp⎜ − ⎟
⎝ Λ⎠
(4)
where Λ is the mean step length and s is the distance of
sediment motion from the pickup point. On the basis of
probability theory, Einstein13) proposed Λ = αd, in which
α is an empirical constant.
The non-equilibrium bed load transport potential rate
by Tsujimoto and Motohashi12) for each sediment size
fraction is written as follows:
qk (x ) =
∞
A3 ∞
d k ∫ p sk ( x − x′)∫ f sk (s )dsdx′
A2 0
x′
(5)
where A2 and A3 are geometric coefficients of sediment
particles.
(2) Equilibrium bed load transport approach
The equilibrium bed load transport formula proposed
by Ashida and Michiue14) can be expressed as:
⎛
τ ⎞⎛
τ
qk = 17 Rgd k3τ *3ek/ 2 ⎜1 − K c *ck 0 ⎟⎜1 − K c *ck 0
⎜
⎟
τ * fk ⎠⎜⎝
τ * fk
⎝
⎞
⎟ (6)
⎟
⎠
where dk is the characteristic diameter of sediment size
fraction k, g is the gravitational constant, R is the relative
density of the sediment (R=ρs/ρ-1), τ*fk is the
dimensionless shear stress, τ*ek is the effective
non-dimensional shear stress, τ*ck0 is the dimensionless
critical shear stress and Kc is a correction factor on the
magnitude of the transport rate for the influence of bed
slope. The dimensionless effective shear stress for each
size fraction can be calculated by using the representative
grain size dk and the shear velocity as follows:
- 721 -
u*2
1
Rgd k ⎛
⎛
⎞⎞
h
⎜ 6 + 1 ln⎜
⎟
⎜ d (1 + 2τ ) ⎟⎟ ⎟
⎜
κ
*m ⎠ ⎠
⎝ m
⎝
2
2
u
u
= * ;τ * fk = * ;τ ck 0 = ζ iτ *cm
Rgd m
Rgd k
zb = Em + Et + N b .Ed + z0
τ *ek =
τ *m
where zb is the bed elevation, Em is the thickness of mixed
Table 1 Cases and conditions for sediment size
fraction concentration calculation
(7)
Cases
A
B
C
D
where h is the local water depth, dm is the mean grain size
of bed material, τ*cm is the dimensionless critical shear
stress for grain size dm, ζi is the coefficient for hiding and
exposure which is common to use the Egiazaroff’s
formular adjusted by Ashida and Michiue14)
2
⎡ log10 (19) ⎤
di
ζi = ⎢
≥ 0.4,
⎥ for
(
)
log
19
d
/
d
d
i
m ⎦
m
⎣ 10
dm
di
ζ i = 0.85
for
≤ 0.4
di
dm
(8)
The slope effect on sediment transport magnitude Kc
can be express as:
Kc = 1 +
1 ⎡⎛
1⎞
∂z ⎤
∂z
⎜1 + ⎟ cos(α ) + sin (α ) ⎥
⎢
μ s ⎣⎝ R ⎠
∂n ⎦
∂s
(9)
where α is the flow direction near the bed and µs is the
static friction coefficient for sediment.
Then, the bed deformation is computed by using the
sediment continuity equation which is
∂z b
1 ∂ ⎛ nk
⎞
+
. ⎜ ∑ qbk ⎟ = 0
∂t 1 − λ ∂x ⎝ k =1 ⎠
(11)
(10)
where zb is the bed elevation and λ is porosity of bed
material.
3.2 The bed layer model
The bed layer model proposed by Ashida et al.5) is
employed for the grain sorting simulation of non-uniform
sediment in our numerical model. In the bed layer
model, bed material is divided into sublayers which are a
mixed layer, a transition layer and a deposited layer.
The mixed layer represents the exchange layer or top
layer containing the bed materials which is active to the
transport process. The mixed layer thickness is assumed
to be constant and equivalent to the size d90 of initial bed
material distribution5). The transition layer acts as a
buffer layer between the mixed layer and the deposited
layer. The thickness of transition layer is a function of
time and streamwise direction and is restricted between 0
< Et ≤ Ed, where Et is the thickness of transition layer and
Ed is the thickness of multiple layers. The deposited
layer is divided into Nb layers in which the thickness of
sublayers is equal to Ed. Therefore, thickness of the
deposition layer is equal to the multiplication of Nb and
Ed.
As to the initial nonuniform bed, the bed elevation is
calculated by
Bed deformation (Δzb)
aggradation
aggradation
degradation
degradation
Conditions
Etn+Δzb≤Ed
Etn+zb>Ed
Etn+Δzb>0
Etn+Δzb≤0
layer, Nb is the total number of sub-layers in the deposited
layer, and z0 is the datum elevation.
In our calculation codes, the nonuniform sediment
transport model is employed to calculate sediment pickup
rate, sediment deposition rate and bed deformation in
each calculated cell.
Then, the concentration of
sediment size fractions can be calculated from the bed
layer model. Details of size fraction concentration
calculation can be expressed in four cases as shown in
Table 1.
4. Experiment
Miwa and Daido15) conducted experiments of
nonuniform sediment.
Both the sand wave
characteristic and sediment sorting inside sand waves
were observed in order to study the interaction between
sediment sorting and formation of sand waves. The
experiment of uniform sediment was also carried out in
order to compare with the nonuniform sediment case.
Experiments were conducted in a straight rectangular
open channel, 6.5 m long, 0.2 m wide and 0.3 m deep.
Four sets of nonuniform sediment were used. The
observed data from five cases of a nonuniform sediment
experimental study are shown in Table 2. Hydraulic
conditions of all experiments are given in the table in
which all parameters were measured after bed forms
reached to their equilibrium stage. In the table, qw is
water discharge per unit width, Ie is the energy slope, u*
is the mean shear velocity, L and Δ are wave length and
wave height of sand dune, respectively.
5. Model validation and sensitivity study
In present study, we have conducted numerical
simulations in order to validate and test the sensitivity of
our morphodynamic model for nonuniform sediment.
The details of studied result can be drawn as follow:
(1) The influence of mean step length on bedform
characteristics and their evolution
With the use of nonequilibrium sediment transport
formula proposed by Tsujimoto and Motohashi12) on
nonuniform sediment, the mean step length is one of the
significant parameter that affects the bed deformation and
bedform geometry. Many researchers have been applied
and proposed various values of mean step length in their
studies. For instant, Wu et al.2,16) used the wave
- 722 -
Table 2 Experimental cases and conditions
Sediment
mean diameter
(cm)
0.070
Ie
(x10-3)
u*
(cm/s)
L
(cm)
Δ
(cm)
350
Water
depth
(cm)
8.21
3.50
3.31
30.70
0.67
Run-2
400
8.87
0.070
4.00
3.69
54.00
1.16
Run-3
400
8.83
0.076
2.50
4.50
42.49
1.03
Run-4
350
8.38
0.073
2.50
4.18
42.45
0.88
Run-5
450
8.86
0.075
2.50
5.08
46.50
0.81
Run no.
qw
(cm2/s)
Run-1
Table 3 The mean step length value for numerical
simulation cases
Case
A
B1
B2
B3
B4
B5
Mean step
length
Dune length
10d
20d
30d
35d
50d
Remark
Van Rijn
Nakagawa et al.18)
Nakagawa et al. 18)
Nakagawa et al. 18)
Nakagawa et al. 18)
Nakagawa et al. 18)
length of sand dunes on the bed as the length of mean
step length in the calculation. From the study of Van
Rijn17), when sand dunes are a dominant bedform, the
mean step length may be taken as a wave length of sand
dune. Based on a large number of reliable flume and
field data, Van Rijn proposed the following expression
for the bedform length:
Ls = 7.3h
(12)
where Ls is the bedform length and h is water depth.
Moreover, the mean step length of nonuniform material
was experimentally investigated by Nakagawa et al.18).
It was found that the distribution of mean step length for
each grain size was approximated by an exponential
distribution. The empirical constant (α) for each grain
size was almost constant and was proposed to be 10-30.
In this study, the mean step length formula proposed
by Van Rijn 17) and Nakagawa et al.18) are employed as
shown in Table 3. We performed 30 cases of numerical
calculation (one case of experiment was tested with six
different mean step length vales).
The simulation results for all cases are shown in Table
4. We found that in the cases of the high value of mean
step length (case A and case B5), the initial perturbation
on bed disappears and no bedform appears with the
increase of time. Whereas in the case of the smallest
value of mean step length (case B1) causes the instability
on bedform evolution, and dunes cannot maintain, in
which small perturbations are found in the last stage.
However, we found that the appropriate range of the
mean step length for our simulation is found to be in the
range of 20-30d, which is comparable with the study of
Nakagawa et al. 18).
Figure 1 shows the instantaneous bed configurations
of numerical simulation of experiment Run-2 where a, b,
c, d, e and f show the simulation results in the cases of
mean step length equal to dune length, 10d, 20d, 30d, 35d
and 50d, respectively. Figure 1a shows simulation
result in the case of mean step length equal to the length
of sand dune. From the calculation result, it is found
that the initial perturbations on the initial bed disappear
and no bedform appears with the increase of time.
Figure 1b shows simulation result in the case of mean
step length equal to 10d. In the beginning of calculated
time, the bedforms are generated but their evolutions are
unstable with the increase of time, in which small
perturbations are found in the last stage. Figure 1c
shows simulation result in the case of mean step length
equal to 20d. The sand dunes and grain sorting are
generated with the increase of time. The sand dunes
reach to the equilibrium state after 1800 seconds which
are similar to the experiment. By using the mean step
length equal to 20d, the numerical model provides a
under predicted value of dune heights and dune lengths
compared with the observed data. Figure 1d shows
simulation result in case of the mean step length equal to
30d. The sand dunes appear and grain sorting is also
generated with the increase time. The sand dunes reach
to the equilibrium stage after 1800 seconds which are
similar to the experiment. With the use of the mean step
length equal to 30d, dune lengths and dune heights are in
good agreement with the observed data in all cases.
Figure 1e shows simulation result in the case of mean
step length equal to 35d.
Among six cases of
experimental study, sand dunes can maintain on the bed
only in the case of Run-2-B4 and small perturbation
appears on the bed in other cases. In the case of
Run-2-B4 the bedform migration becomes equilibrium
after 3000 seconds which longer than the experiment.
Figure 1f shows simulation result in the case that the
mean step length is equal to 50d.
The initial
perturbations disappear and no bedforms appear with the
increase time for all experimental cases.
With the use of the nonequilibrium sediment transport
approach, Eulerian stochastic formula proposed by
Tsujimoto and Motohashi12), the model ability on
bedform geometry prediction depends on the value of the
mean step length. The smaller mean step length
provides the smaller wave length of sand dune. The
- 723 -
Table 4 Simulation results of various mean step length
cases
Cases
Run-1-A
Run-1-B1
Run-1-B2
Run-1-B3
Run-1-B4
Run-1-B5
Run-2-A
Run-2-B1
Run-2-B2
Run-2-B3
Run-2-B4
Run-2-B5
Run-3-A
Run-3-B1
Run-3-B2
Run-3-B3
Run-3-B4
Run-3-B5
Run-4-A
Run-4-B1
Run-4-B2
Run-4-B3
Run-4-B4
Run-4-B5
Run-5-A
Run-5-B1
Run-5-B2
Run-5-B3
Run-5-B4
Run-5-B5
Wave
length
(cm)
20.02
30.40
20.03
27.57
56.80
19.96
34.83
19.97
28.71
29.04
40.33
-
Wave
height
(cm)
0.59
0.89
0.66
0.93
0.91
0.58
0.87
0.51
0.60
0.90
1.02
-
Remark
Flat bed
Perturbation bed
Sand dune
Sand dune
Small perturbation
Flat bed
Flat bed
Perturbation bed
Sand dune
Sand dune
Sand dune
Flat bed
Flat bed
Perturbation bed
Sand dune
Sand dune
Small perturbation
Flat bed
Flat bed
Perturbation bed
Sand dune
Sand dune
Small perturbation
Flat bed
Flat bed
Perturbation bed
Sand dune
Sand dune
Small perturbation
Flat bed
Run-x-yy: x refers to experimental case
yy refers to mean step length case
a. Shows simulation results of Run-2-A
(mean step length= dune length)
b. Shows simulation results of Run-2-B1
(mean step length= 10d)
c. Shows simulation results of Run-2-B2
(mean step length= 20d)
d. Shows simulation results of Run-2-B3
(mean step length= 30d)
e. Shows simulation results of Run-2-B4
(mean step length= 35d)
f. Shows simulation results of Run-2-B5
(mean step length= 50d)
Fig. 1 The instantaneous bed configurations of
numerical simulation result of experiment Run-2
- 724 -
100
layer I (observed)
90
layer II (observed)
Percent Finer
80
layer III (observed)
70
layer I (computed)
60
layer II (computed)
50
layer III (computed)
40
30
20
10
0
0.01
0.10
1.00
Grain diameter(cm)
a. The comparison of grain size distributions (Run-2-B3) where the locations of measured mixture are shown as a
small figure.
2.0
Run-1-B3
Run-2-B3
Run-3-B3
Run-4-B3
Run-5-B3
50
Run-1-B3
Run-2-B3
Run-3-B3
Run-4-B3
Run-5-B3
-30%
40
Experimental wave height (cm)
Experimental wave length (cm)
60
+30%
30
20
1.5
-30%
+30%
1.0
0.5
10
0.0
0
0
10
20
30
40
Simulated wave length (cm)
50
0.0
60
0.5
1.0
1.5
Simulated wave height (cm)
2.0
b. The comparison between the experimental results and c. The comparison between the experimental results and
the simulated results of the wave height of sand dunes.
the simulated results of the wave length of sand dunes.
Fig. 2 The comparison of sand dune geometries between the experimental results and the simulated results (the mean
step length = 30d)
model results are comparable with the study of
Yamaguchi and Izumi19). They performed a linear
stability analysis by the use of the stochastic model with
two-dimensional Reynolds equation on dune formation of
uniform sediment. They found that the case of small
value of mean step length, sand dunes can be generated in
the large range of unstable wave number therefore the
bed is composed of many sizes of wave length and then
causes a perturbation bed. The unstable area of dune
formation is large. In the case of high value of mean
step length, the range of unstable wave number for
generating instability becomes small therefore only large
wave length of sand dune can be generated and
maintained on the bed. The unstable area of dune
formation becomes small.
Moreover, some recent
studies on numerical model of the uniform sediment
using the Eulerian stochastic formula, Giri and Shimizu20)
and Toyama et al.21), showed that the simulation results of
the bedform evolution are sensitive to the value of mean
step length. However, the influence of mean step length
on dune evolution in the case of nonuniform sediment
needs the further study in order to clarify their effects on
dune formation.
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Fig. 3 Instantaneous feature of numerical simulation
result with equilibrium sediment transport model (top)
and nonequilibrium sediment transport model (bottom)
Fig. 4 Simulation results with random field shape of
initial perturbation (top) and cosine shape of initial
perturbation (bottom)
Fig. 5 Instantaneous simulation results with 2 m.
domain length (top) and 4 m. domain length (bottom)
Among six cases of various mean step length which
are applied to all experimental cases, it is found that the
numerical model is in best agreement with the
experiments in geometry and grain sorting when mean
step length is equal to 30d as shown in Figure 2.
Figure 2 shows the comparison between the simulated
results and the experimental results of sand dune
geometry and grain sorting inside the sand dunes using
the mean step length = 30d.
Figure 2a show the comparison between the simulated
results and the experimental results on grain sorting at
various locations inside the sand dunes of the case
Run-2-B3, and a small figure shows three locations of the
measured mixture as Layer I, II and III. Layer I denotes
the mixture on the upstream part of dune crests, Layer II
denotes the mixture in the trough of dunes, and Layer III
denotes the mixture in the substrate layer. From the
experimental results, it was found that Layer I provides
the finest mixture among three layers. Layer II which is
coarser than Layer I and III because coarse grains was
deposited at the lee side, whereas the grain size
distribution in Layer III shows no significant change
because it does not strongly participate with the sediment
transport and bed evolution.
Comparing with the
simulated results, the grain size distribution of bed
material inside the sand dunes in layer II and layer III are
reproduced very well by our numerical model.
Figure 2b shows the comparison between the
simulated results and the experimental results of the wave
length of sand dunes. The results show that the
numerical model provides the under predicted wave
lengths of sand dunes with the discrepancy 30% in most
cases (3 in 5 cases). We found that the under predicted
wave lengths of sand dunes are affected by the water
depth. The present model provides the under estimated
flow depth, therefore causes the under prediction of wave
length. The model shows a fair agreement on the wave
length prediction. However, some improvement on
flow model is needed in order to well predict the flow
depth and the wave length of sand dunes.
Figure 2c shows the comparison of the wave heights
of sand dunes between the simulated results and the
experimental results. It is found that the numerical
model shows a satisfactory agreement on the wave height
predictions with the discrepancy ±30% in most cases.
(2) Influence of sediment transport model on bedform
configuration simulation
In this study, we attempt to evaluate the performance
of our morphodynamic model for nonuniform sediment
by using the different sediment transport approach. The
equilibrium sediment transport approach, a bed load
transport formula proposed by Ashida and Michiue14), is
employed in the nonuniform sediment transport module.
The instantaneous features of simulation result of Run-2
are depicted in Figure 3. With the use of the bed load
transport formula proposed by Ashida and Michiue14), it
is showed that the perturbation grows rapidly and the
sand dune cannot be generated on the bed. The grain
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sorting simulation appears to be incompatible with the
physical observation.
The flow depth is also
overestimated. Whereas, the simulation results using
the nonequilibrium sediment transport model show a
satisfactory agreement in bedform geometry and grain
sorting prediction5). From the simulation result, it can
concluded that the use of nonequilibrium sediment
transport model is an appropriate approach for simulating
micro-scale bedform dynamics, because it can reproduce
the phase lag between bed shear stress and the particle
transport by incorporating the step length.
(3) Model sensitivity to initial perturbation
In the computational method, we put small
perturbations on the initial bed condition and then allow
it to grow using our simulation technique.
Two
different perturbation shapes, random shape and cosine
shape, are used in our model. It is necessary to assess
the effect of the initial perturbation shape on bedform
evolution. We test both types of initial perturbation
shape in experiment case of Run-3. The instantaneous
bedform features from simulation result are illustrated in
Figure 4. Some small alteration can be noticed in size of
bedform. However, no any significant quantitative
distinction can be made between these two of cases.
(4) Model sensitivity to domain length
The periodic boundary condition was employed in our
calculation in order to reduce the computational time. It
is necessary to assess the effect of calculation domain
length in order to confirm the periodic phenomenon.
We used two different domain lengths with all
experimental cases, namely 2m and 4m.
The
quantitative comparison of an instantaneous bedform
configuration of Run 3 is shown in Figure 5. The
comparison result shows that the numerical model is
sufficient to simulate the experimental cases. However,
the full length of flume should be investigated in the
further study.
6. Conclusions
The numerical simulation studies were conducted in
order to investigate on the morphodynamic model for
nonuniform sediment.
From this study, some
conclusions can be drawn as follows:
1) The appropriate mean step length for nonuniform
sediment is found to be in the range of 20-30d. By
using the mean step length equal 30d, it is found that our
morphodynamic model is in good agreement of bedform
geometry and grain sorting predictions.
2) With the use of nonequilibrium sediment transport
model, Eulerian stochastic formula by Tsujimoto and
Motohashi, the model ability on dune formation
prediction appears to be depending on the mean step
length.
3) The model performance on sand dune dynamics and
grain sorting simulation differs based on the sediment
transport model. A nonequilibrium sediment transport
model seems to be more appropriate than an equilibrium
model on micro-scale of bedform simulation. Although
the equilibrium sediment transport model proposed by
Ashida and Michiue seems adequately to reproduce the
bed deformation on meso-scale of bedform simulation,
the model shows poor results on micro-scale of bedform
simulation.
4) The morphordynamic model appears to be insensitive
with the initial perturbation.
ACKNOWLEDGMENT: We gratefully acknowledge
Assoc. Prof. Dr. Hiroshi Miwa for providing unpublished
experimental data.
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(Received: April 9, 2009)